Here's an outline and a summary of what's introduced in this tutorial
Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IITJEE
Main or Advanced/AIEEE
, and anyone else who needs this Tutorial as a reference!
After studying the topic you might also be interested in attempting the following 2 Quizzes related to Complex Numbers.
After reading this tutorial you might want to check out some of our other Mathematics Quizzes as well.
Complex Numbers
Basics of Complex Numbers, Real and Imaginary Parts, Iota
 Complex Numbers is the largest and the complete set of numbers, consisting of both real and unreal numbers.
 A complex number is of the form
a+ib, where a and b are both real numbers, and i is called “iota” and is defined to be the square root of 1. That is, i=√1. Also, i2=1.
 A complex number is usually denoted by the letter ‘z’.
 ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number.
 The notion of complex numbers increased the solutions to a lot of problems. For instance, had complex numbers been not there, the equation x2+x+1=0 had had no solutions.
Because of ‘i', we can now extend the square root to be defined for negative numbers as well. If y is a negative number, then y can be written as y where y is the absolute value of y. Then, the square root of y is defined to be equal to be
y=i√y
Algebra of Complex Numbers, Conjugate, Modulus and Argument, Euler's Form
 Complex Numbers have different algebraic laws as well. Let z1=a+ib,z2=c+id.
 z1+z2=(a+c)+i(b+d)
 z1z2=(ac)+i(bd)
 z1.z2=(acbd)+i(ad+bc)
 z1/z2=((ac+bd)+i(adbc))/(c2d2)
 1/z1=(aib)/(a2b2)
 Conjugate : Conjugate of a complex number z=a+ib is defined as
z=aib.
 Modulus of z is written as z and is defined to be equal to √(a2+b2)
 Thus, z.Conjugate(z)=z2.
 Commutative, Associative, and Distributive Laws of algebra apply to Complex Algebra as well.
 Complex Numbers can be represented pictorially on a plane known as Argand Plane.
 The plane consists of a Real Axis, and an Imaginary Axis.
 The angle that z makes with the real axis (counterclockwise) is called the argument of z, and the length of line segment joining z and the origin is equal to z, also sometimes written as ‘r’.
 Different representations of z,
z=a+ib
z=r(cosθ + i sinθ) [Modulus Argument Representation]
z=reiθ [Euler’s Form]
 Observe that, eiθ=r(cosθ + i sinθ)
 Multiplication and Division using using Euler’s Form:
Let z1=r1eiθ1, z2=r2eiθ2
z1z2=r1r2ei(θ1+θ2)
z1/z2=(r1/r2)ei(θ1θ2)
 Conversion between various forms.
r=z;
Let α=tan1(b/a); then θ=α,πα,απ,θ depending upon z being in first, second, third or fourth quadrant respectively. This is called the principal argument.
 Approaches to solving questions:
 Expand every complex number as a real and imaginary part.
 Use Euler’s form for every Complex Number
 Use Algebraic Identities, and solve in general, keeping in mind that complex numbers have a real and an imaginary part.
Complex Numbers as Free Vectors
 Another approach to dealing with Complex Numbers is to treat them like free vectors. Free Vectors are vectors that can be moved around without changing direction, in other words, that do not really matter upon the origin. The approach that was used so far was treating complex number as a point.
 Any comple number such as 2+3i can be any one of the infinite number of vectors
 eiθ is multiplied with the vector which is to be rotated, and rotates the vector by “θ” anticlockwise. Clockwise rotation is obtained by multiplying with eiθ
 m, n are unit vectors; and we have neiθ=m, m eiθ=n
Here's a quick look at some of the questions we will solve
 What is the value of (1+i)(1+i2)(1+i3)(1+i4)?
 If (1+i)(1+2i)(1+3i). . .(1+ni)=a+ib, then what is 2X5X10X…X(1+n2)?
 If (x+iy)1/3=a+ib; then x/a+y/b=?
 Find the sum of i+i2+i3+…upto 1000 terms.
 If z=2, find the area of the triangle formed by z,iz,z+iz.
 If x=5+2√4, find the value of x4+9x3+35x2x+4.
 Find the square root of 724i.
 Given that cosα+cosβ+cosγ=0 and sinα+sinβ+sinγ=0, prove that sin3α+sin3β+sin3γ=3sin(α+β+γ)
 If cosα+2cosβ+3cosγ=0 and sinα+2sinβ+3sinγ=0. Show cos3α+8cos3β+27cos3γ=3sin(α+β+γ).
 Given that 1,α1,α2,…,αn1 are nth roots of unity, Show that (1 α1)(1 α2)…(1 αn1)=n.
 Show that sin(π/n)sin(2π/n)…sin((n1)π/n)=n/2n1
Complete Tutorial Document with Examples and Solved Problems (There is an MCQ Quiz after this)
MCQ Quiz #1
Companion MCQ Quiz/Worksheet #1 for Complex Numbers (The Basics)  test how much you know about the topic. Your score will be emailed to you at the address you provide.
MCQ Quiz: Complex Numbers The BasicsGoogle Spreadsheet Form
MCQ Quiz/Worksheet #2: More on Complex Numbers
Read the questions in the document below, and fill up the answers in the answer submission form below it.
Your score will be emailed to you
Questions: More on Complex NumbersGoogle Document
Answer Submission Form for MCQ Quiz #2
Answer Submission Form for MCQ Quiz #2: More on Complex NumbersGoogle Spreadsheet Form
You might like to take a look at our other algebra tutorials:
Introduction to Complex Numbers
Introduction to Complex Numbers and iota. Argand plane and iota. Complex numbers as free vectors. Nth roots of a complex number. Notes, formulas and solved problems related to these subtopics. 
The Principle of Mathematical Induction Introductory problems related to Mathematical Induction. 
Quadratic Equations
Introducing various techniques by which quadratic equations can be solved  factorization, direct formula. Relationship between roots of a quadratic equation. Cubic and higher order equations  relationship between roots and coefficients for these. Graphs and plots of quadratic equations. 
Quadratic Inequalities
Quadratic inequalities. Using factorization and visualization based methods. 
Series and Progressions
Arithmetic, Geometric, Harmonic and mixed progressions. Notes, formulas and solved problems. Sum of the first N terms. Arithmetic, Geometric and Harmonic means and the relationship between them. 



